Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682340 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36122 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(10,13)(11,15)(14,16)(17,18,21,22)(19,20,23,24);(1,3)(2,5)(4,7)(6,8)(18,22)(19,23);(2,6)(3,7)(9,10,11,13,12,14,15,16)(17,19,20,22,21,23,24,18);(17,20,21,24)(18,19,22,23);(3,7)(5,8)Polycyclic presentation:
2^9|a^2=f,c^2=fgh,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6670776);